Theorem carden2b 9863

Description: If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 9862 are meant to replace carden 10445 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)

Assertion

Ref	Expression
carden2b	⊢ (𝐴 ≈ 𝐵 → (card‘𝐴) = (card‘𝐵))

Proof of Theorem carden2b

Step	Hyp	Ref	Expression
1		cardne 9861	. . . . 5 ⊢ ((card‘𝐵) ∈ (card‘𝐴) → ¬ (card‘𝐵) ≈ 𝐴)
2		ennum 9843	. . . . . . . 8 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ dom card ↔ 𝐵 ∈ dom card))
3	2	biimpa 476	. . . . . . 7 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → 𝐵 ∈ dom card)
4		cardid2 9849	. . . . . . 7 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
5	3, 4	syl 17	. . . . . 6 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐵) ≈ 𝐵)
6		ensym 8928	. . . . . . 7 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴)
7	6	adantr 480	. . . . . 6 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → 𝐵 ≈ 𝐴)
8		entr 8931	. . . . . 6 ⊢ (((card‘𝐵) ≈ 𝐵 ∧ 𝐵 ≈ 𝐴) → (card‘𝐵) ≈ 𝐴)
9	5, 7, 8	syl2anc 584	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐵) ≈ 𝐴)
10	1, 9	nsyl3 138	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → ¬ (card‘𝐵) ∈ (card‘𝐴))
11		cardon 9840	. . . . 5 ⊢ (card‘𝐴) ∈ On
12		cardon 9840	. . . . 5 ⊢ (card‘𝐵) ∈ On
13		ontri1 6341	. . . . 5 ⊢ (((card‘𝐴) ∈ On ∧ (card‘𝐵) ∈ On) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴)))
14	11, 12, 13	mp2an 692	. . . 4 ⊢ ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴))
15	10, 14	sylibr 234	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐴) ⊆ (card‘𝐵))
16		cardne 9861	. . . . 5 ⊢ ((card‘𝐴) ∈ (card‘𝐵) → ¬ (card‘𝐴) ≈ 𝐵)
17		cardid2 9849	. . . . . 6 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
18		id 22	. . . . . 6 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ≈ 𝐵)
19		entr 8931	. . . . . 6 ⊢ (((card‘𝐴) ≈ 𝐴 ∧ 𝐴 ≈ 𝐵) → (card‘𝐴) ≈ 𝐵)
20	17, 18, 19	syl2anr 597	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐴) ≈ 𝐵)
21	16, 20	nsyl3 138	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → ¬ (card‘𝐴) ∈ (card‘𝐵))
22		ontri1 6341	. . . . 5 ⊢ (((card‘𝐵) ∈ On ∧ (card‘𝐴) ∈ On) → ((card‘𝐵) ⊆ (card‘𝐴) ↔ ¬ (card‘𝐴) ∈ (card‘𝐵)))
23	12, 11, 22	mp2an 692	. . . 4 ⊢ ((card‘𝐵) ⊆ (card‘𝐴) ↔ ¬ (card‘𝐴) ∈ (card‘𝐵))
24	21, 23	sylibr 234	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐵) ⊆ (card‘𝐴))
25	15, 24	eqssd 3953	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ dom card) → (card‘𝐴) = (card‘𝐵))
26		ndmfv 6855	. . . 4 ⊢ (¬ 𝐴 ∈ dom card → (card‘𝐴) = ∅)
27	26	adantl 481	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐴) = ∅)
28	2	notbid 318	. . . . 5 ⊢ (𝐴 ≈ 𝐵 → (¬ 𝐴 ∈ dom card ↔ ¬ 𝐵 ∈ dom card))
29	28	biimpa 476	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐴 ∈ dom card) → ¬ 𝐵 ∈ dom card)
30		ndmfv 6855	. . . 4 ⊢ (¬ 𝐵 ∈ dom card → (card‘𝐵) = ∅)
31	29, 30	syl 17	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐵) = ∅)
32	27, 31	eqtr4d 2767	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐴) = (card‘𝐵))
33	25, 32	pm2.61dan 812	1 ⊢ (𝐴 ≈ 𝐵 → (card‘𝐴) = (card‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ⊆ wss 3903  ∅c0 4284   class class class wbr 5092  dom cdm 5619  Oncon0 6307  ‘cfv 6482   ≈ cen 8869  cardccrd 9831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6310  df-on 6311  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-er 8625  df-en 8873  df-card 9835

This theorem is referenced by:  card1  9864  carddom2  9873  cardennn  9879  cardsucinf  9880  pm54.43lem  9896  nnadju  10092  nnadjuALT  10093  ficardun  10095  ackbij1lem5  10117  ackbij1lem8  10120  ackbij1lem9  10121  ackbij2lem2  10133  carden  10445  r1tskina  10676  cardfz  13877